If the speaker wasn't ambivalent about priests, preachers, and clergymen, we'd think he was a preacher himself. The poem sounds like a really long, deeply passionate sermon, and the audience is America as a whole.
The sound of a Whitman poem – especially this Whitman poem – couldn't be more recognizable. The most important audible feature of the poem is what it does to your breath. When you read it aloud, you tend to take a breath after each line. That's because the lines are longer than most poems you've probably read. A lot of the lines are just run-on sentences and fragments crammed together. For example, "Out of the dimness opposite equals advance – always substance and increase, always sex" (section 2). If you read the entire poem aloud, first of all you'd be a hero in our book, but, second, you'd also be about as winded as if you had just run five miles. If you've ever heard a hypnotic preacher, you'll notice that they often get tired and sweaty on the pulpit. It's hard to keep people's attention for that long! In the same way, Whitman is truly giving it all he's got here.
Another tactic used by preachers is repetition. Think of Martin Luther King, Jr.'s speech "I Have a Dream." King, of course, was a preacher, and in that speech he repeats the phrase "I Have a Dream" over and over again in a spellbinding way. Much of the speech is a list, or "catalogue," of King's dreams for society. Whitman is the poet who uses repetition of phrases in successive lines (called anaphora) and long lists (called catalogues). Take, for example, this list of things that the speaker "knows" for certain:
Swiftly arose and spread around me the peace and knowledge that
pass all the argument of the earth;
And I know that the hand of God is the promise of my own,
And I know that the spirit of God is the brother of my own;
And that all the men ever born are also my brothers, and the
women my sisters and lovers; (section 5)
This particular list only goes on for four more lines, but there are some that go on for many, many lines, like the list of people and places in Section 33.
Two other tricks of the trade used by our speaker are rhetorical questions and apostrophe. In a sermon, a preacher uses rhetorical questions to anticipate an argument or to introduce a subject by pretending to respond to a question about it. Whitman does exactly the same thing with questions like, "Have you felt so proud to get at the meaning of poems?" (section 5). In that case, the argument he's responding to is, "I feel proud because I understand poems."
As for apostrophe, the term just means talking to someone who can't respond. Preachers might talk to the "poor and oppressed" or to "sinners" or some other group. Politicians often address "My fellow Americans." The sound of "Sound of Myself" is greatly influenced by the speaker's repeated attempts to start side-conversations, as if he were pointing to someone in a great crowd and saying, "You there! I've got something to say to you." Who, me?
Whitman generally avoids the fancy language and sentence structures of the traditional poetry of his day, but his tone is more like a formal speech than a plain conversation. He still knows how to turn a phrase. Think, for example, of the memorable passage when he praises his soul, says he wants to hears its "valved voice," and remembers when his soul "plunged [its] tongue to my bare-stript heart." From start to finish, "Song of Myself" is full of such astonishingly beautiful and highly eloquent passages. If Whitman was a preacher, we wouldn't miss his sermons for the world.